From now on we fix \(K \in \mathbb {R}\)  and a finite \(N<\infty \) .
Thanks to recent quick developments on the study of \(\operatorname{RCD}(K, N)\)  spaces, most of the well-known properties on Ricci limit spaces can be covered by the \(\operatorname{RCD}\)  theory. For example, it is proved in [1]} that the essential dimension, denoted by \(\dim (X, \mathsf {d}, \mathfrak {m})\) ,We know that \(\dim _{\mathcal {H}}(X, \mathsf {d}) \ge \dim (X, \mathsf {d}, \mathfrak {m})\)  holds. See the final section. It is also conjectured that \(\dim _{\mathcal {H}}(X, \mathsf {d}) = \dim (X, \mathsf {d}, \mathfrak {m})\)  holds. See also [2]}. whose definition is the unique \(k\)  such that the \(k\) -dimensional regular set has positive \(\mathfrak {m}\) -measure, is well-defined. This gives a generalization of a result proved in [3]} to \(\operatorname{RCD}(K, N)\)  spaces. See also [4]}.
